inapproximabilitytuloksiin

Inapproximability results are theoretical statements in computer science that establish fundamental limits on the efficiency of approximation algorithms for certain computational problems. These results often demonstrate that if a problem can be approximated within a certain factor, then a famous computationally hard problem, like P versus NP, would also be solvable efficiently. The typical form of an inapproximability result is that for a specific problem P, there is no polynomial-time approximation scheme (PTAS) unless P=NP, or that it is NP-hard to approximate P within a factor of c, for some constant c.

These results are crucial because they guide researchers away from attempting to find efficient approximation algorithms for problems that are proven to be inherently difficult to approximate. Instead, they suggest focusing on developing algorithms with provable approximation ratios that are as close as possible to the inapproximability bound, or exploring alternative computational models. The proofs of inapproximability results often involve complex reductions from known NP-hard problems to the problem in question, showing that an efficient approximation for the latter would imply an efficient solution for the former.